3.13. Arbitrary Lagrangian-Eulerian

Aria supports solving general transport equations on grids with arbitrary grid motion using an Arbitrary Lagrangian-Eulerian formulation. The following is a brief introduction based on a series of detailed lecture notes [32].

Consider a region undergoing arbitrary mesh motion described by the mapping $\mathbf{x} = \hat{\Phi}\left(\mathbf{\chi}, t\right)$ , where $\mathbf{x}$ is a spatial coordinate and $\mathbf{\mathbf{\chi}}$ is the referential coordinate. For this region, conservation of mass written with respect to the current spatial configuration is

$\int_{\Omega_x = \hat{\Phi}\left(\Omega_{\chi}\right)} \hat{J}^{-1} \frac{\partial \hat{J} \rho}{\partial t} \bigg|_{\mathbf{\chi}} + \nabla \cdot \left(\rho \mathbf{c} \right) = 0.$

Here, $\rho$ is the density, $\mathbf{c} = \mathbf{v} - \mathbf{v}_m$ is the convective velocity with $\mathbf{v}$ the material velocity and $\mathbf{v}_m$ the mesh velocity. The time derivative here is with $\mathbf{\chi}$ held fixed i.e., it is the referential time derivative. The gradient operator $\nabla$ , unless otherwise stated, is with respect to the spatial coordinate $\mathbf{x}$ . The determinant of the mesh Jacobian $\hat{J}$ is defined as

$\hat{J} = \det \mathbf{\hat{F}},$

where the mesh Jacobian is $\mathbf{\hat{F}} = \nabla_{\mathbf{\mathbf{\chi}}} \hat{\Phi}$ . Using the following relation for referential time derivative of the determinant of the mesh Jacobian

$\frac{\partial \hat{J}}{\partial t}\bigg|_{\mathbf{\chi}} = \hat{J} \nabla \cdot{\mathbf{v}_m},$

the strong form, with the advective term written in terms of the divergence of the relative mass flux is

$\frac{\partial \rho}{\partial t} \bigg|_{\chi} + \nabla \cdot \left(\rho \mathbf{c} \right) + \rho \nabla \cdot \mathbf{v}_m = 0.$

The equivalent strong form with an advective term written in non-relative mass flux divergence form is

$\frac{\partial \rho}{\partial t} \bigg|_{\chi} + \nabla \cdot \left(\rho \mathbf{v} \right) -\mathbf{v}_m \cdot \nabla \rho = 0.$

Note that in either form, the last term that appears is a mesh motion correction term. For the relative mass flux formulation, this term corrects the mass term, whereas for the non-relative form it corrects both the mass term and advective term i.e., boundary mesh fluxes are not explicitly included.

Lastly, a third form is the so called advective form, i.e.,

$\frac{\partial \rho}{\partial t} \bigg|_{\chi} + \mathbf{c} \cdot \nabla \rho + \rho \nabla \cdot \mathbf{v} = 0.$

Given that the equivalent conservation of mass with respect to a material particle is

$\frac{D \rho}{D t} = -\rho \nabla \cdot \mathbf{v},$

we can obtain the following ALE relation between the material time derivative of some quantity (here the density) and the referential time derivative of the quantity

$\frac{D \rho}{D t} = \frac{\partial \rho}{\partial t}\bigg|_{\mathbf{\chi}} + \mathbf{c} \cdot \nabla \rho,$

This ALE relation can be used to derive advective forms for other equations e.g., for the energy equation

$\rho C_p \dfrac{D T}{Dt} = \rho C_p \left(\frac{\partial T}{\partial t}\bigg|_{\mathbf{\chi}} + \mathbf{c} \cdot \nabla T\right).$

Depending on the equation, Aria supports either one or more of these forms.